As I understand it the Schengen agreement between EU countries allows free passage between the collaborating countries without the need to show passports. The UK does not participate as we don't trust the border controls of other countries. A US citizen travelling to a Schengen country needs to have the correct permit / Visa / whatever to visit that country and then will be free to visit all other Schengen countries.
No, US citizens do not normally need a Schengen visa. They are normally allowed to stay for 90 days as a tourist without a visa.
You could spend only 90 days in the entire Schengen region. For a non-Schengen citizen, the Schengen region is like one nation from the perspective of visa requirements.
Citizens of EU and Schengen countries (including Ireland) have the right to come and go in the EU and Schengen area without restriction. No visas are required, and there is no time limits on the stay.
British citizens are required visa for entering Vietnam.
Schengen visas are obtained from the embassy of the country you are first going to visit in the Schengen area.
You need to apply to the embassy of the first country you intend to visit in the Schengen Area.
Is the question: what countries does the US require a visa from? If so, the answer is that the US requires a visa from every country on Earth EXCEPT those member countries that are part of the Visa Waiver Program (VWP). However, even in these countries, those people with criminal records will STILL be required to obtain a visa.
Switzerland is a part of the Schengen agreement which allows US citizens to travel to the country without a visa. The stay must be under 90 days after which a visa is required.
If you leave the Schengen area and go to a non-Schengen country, and then re-enter the Schengen area later, the "clock" starts again from zero. If you do not need a Schengen visa, this is an advantage. If you do need a visa, it is a disadvantage as you need to get a multiple-entry visa. So, if you plan in a trip outside Schengen in the middle of your travels, you can get 2 x 90 days. For example, if you are in France, you could do a trip to London, as the UK is not in Schengen.
All foreign national are required to have a visa to enter the US. Even American citizens are required to have a passport to re-enter the country.
1) A green card does not mean anything outside the USA. It is your nationality (citizenship) that determines if you need a visa. 2) The UK is not part of the Schengen Area. A Schengen visa is never valid for the UK. You need to ask the UK embassy in the USA (or look on their website) to find out if someone of your nationality requires a transit visa for the UK.
Yes US citizens need a visa.